The following result extends, in particular, [1]} for a larger class of spaces \(\mathbb {E}\)  assuming that, in addition, \(a\)  is in the \(M_3\) -class.
For symbols which arise from particular types of evolution equations, the \(\widetilde{\mathfrak {M}}_1\) -condition (respectively, the \(\mathfrak {M}_1\) -condition) implies the \(\widetilde{\mathfrak {M}}_\gamma \) -condition (respectively, the \(\mathfrak {M}_\gamma \) -condition) for every \(\gamma \in \mathbb {N}\) ; see, e.g. Propositions REF  and REF .
